Weighted mean problems

Keep track of totals, and the average follows. A stork averaging 44.5 km over 20 days has flown 890 km. After 55 km on day 21, the average is (890 + 55)/21 = 45 km.

In a mixture, every concentration uses the whole volume. 600 mL with 5% salt and 300 mL with 8% sugar make 900 mL with 3.33% salt and 2.67% sugar: each ingredient is now spread through all 900 mL.

Diluting keeps the amount of the ingredient fixed. 10 L of 8% pani holds 0.8 L of spice. For 6%, it needs 0.8/0.06 ≈ 13.33 L in total, so add 3.33 L of water.

Averages can reveal group sizes. Langurs: males average 16.5 kg, females 13.8 kg, and all 60 average 14.925 kg. The overall average is closer to 13.8, so there are more females. Solving gives 25 males and 35 females.

Removing values that average the same as their group leaves the average unchanged. Taking out two males of 16.9 and 16.1 kg (average 16.5) leaves the males' average at 16.5.

A mixture always lands between its parts. From 34% and 0.001% water you can reach 0.01%, but you can never go below 0.001%.